Here is the under hand throw with single windup. From the ready the sling is swung forward over the head, behind the back, under the arm, and then the release occurs anywhere from 6 to 3 o'clock in front of the body, depending on how far you want to throw. . It can also be done just by whipping up straight from the ready, without a windup.
Less lethal projectiles than rocks are water balloons, tennis or racquet balls that have been filled with water to increase their weight. This is done with a syringe. The tennis balls self seal, but the racquet balls will require a bit of silicone sealer to keep the water from leaking out.